Invariants of Broken Discrete Symmetries

The parity and Bloch theorems are generalized to the case of broken global symmetry. Local inversion or translation symmetries in one dimension are shown to yield invariant currents that characterize wave propagation. These currents map the wave function from an arbitrary spatial domain to any symmetry-related domain. Our approach addresses any combination of local symmetries, thus applying, in particular, to acoustic, optical, and matter waves. Nonvanishing values of the invariant currents provide a systematic pathway to the breaking of discrete global symmetries.

Figure 1

Different types of symmetry under inversion ($\mathrm{\Pi }$) through $\alpha$ or translation ($T$) by $L$, mapping a domain $\mathcal{D}$ to $\overline{\mathcal{D}}$: (a) global symmetry, (b) nongapped local symmetry, (c) gapped local symmetry, and (d) complete local symmetry.